Precalculus – 7.4 Notes

Powers and Roots of Complex Numbers

De Moivre's Theorem

If ( )cos sinz r θ i θ= + is a complex number and n is any positive integer, then

( )cos sinn nz r nθ i nθ= +

Examples:

a) Simplify ( )6

1 .i+ b) Simplify ( )4

3 .i−

Roots of a Complex Number

How many square roots does 4 have?

How many square roots does 9− have?

How many sixth roots does 1 have?

It turns out that 1 has 6 sixth roots, and they

are distributed evenly around the complex plane.

The complex number a bi+ is an nth root of the complex number z if ( ) .n

a bi z+ =

For any positive integer n, the complex number ( )cos sinz r θ i θ= + has exactly n distinct nth roots given by

the expression ( )1 360°cos sin where n θ k

r α i α αn

++ = for 0,1, 2,..., 1.k n= −

In radians, the roots are given by ( )1 2cos sin where n θ kπ

r α i α αn

++ = for 0,1, 2,..., 1.k n= −

The first of the n roots has an argument of ,θ

n and the other roots are spaced

360

n

�

apart.

(The circle is divided evenly into n pieces.)

Examples:

a) Find all of the fourth roots of the complex number 8 8 3.i− −

b) Find all the cube roots of –125.

c) Find all complex solutions to 3 27 0.x i− =